Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ IL_{2} \& IL_{12}$$\end{document}IL2&IL12) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$IL_{2} IL_{12}$$\end{document}IL2IL12Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

www.nature.com/scientificreports/ between the cell populations and the chemotherapeutic medications 38 .In order to explore the impact of body mass, estrogen, and the immune system on the development of cancer cells, authors provide a mathematical model of fat, estrogen, and breast cancer 39 .After first exposing patients to chemotherapy and virotherapy alone, and then both at once, the authors examine the behaviors displayed by a modified mathematical model that depicts interactions between immune cells, uninfected tumor cells, infected tumor cells, and normal cells 40 .The goal of the authors' work is to use the Caputo and Caputo-Fabrizio fractional operators to explore the dynamics of a nutrient-plankton system 41 .The authors 42 investigate the function of M2 macrophages' saturation response by addressing a tumormacrophage interaction model.The objective of the study conducted by the authors is to examine the effects of discrete-time delay on the immune response to tumor development, excess estrogen, and the source rate of immune cells in a model of breast cancer 43 .
To study the effects of different kinds of immune cells on tumor cells, many models have been developed.Nevertheless, the effect of dendritic cells on tumor cells has only been investigated in a small number of models.Furthermore, surgical techniques have not been explored as a feasible course of treatment in previous talks.On the other hand, our method incorporates both surgery and chemotherapy into our course of care.Furthermore, the authors who came before them did not clearly define the beginning of this therapeutic period.Our work, on the other hand, sets itself apart by presenting a mathematical model of non-small-cell lung cancer that includes the possible therapeutic approaches of chemotherapy and surgery and shows the combined effect of both.Additionally, we offer recommendations for the optimal approach among the various combinations involving surgery and various doses of chemotherapy.
Here, the newly introduced fractional derivatives in the analysis and numerical modelling of the lung cancer have been employed.Lung cancer is among the most dangerous types of cancer which are prevalent in human life because it has the highest tendency.This is done through establishing that the proposed system exists and has a certain set of properties, checking the validity of the solution system quantitatively.More so, by applying the Atangana Blaneao derivative, it is possible to discuss the dynamics of a mathematical model in real life.Lastly, numerical models are employed to enhance and support the biological studies conducted in the experiments.With regards to the particular importance of the aforementioned aspects, one would like to focus our analysis of these basic questions on a model developed specifically to illustrate the dynamics that are evident in lung cancer and the shortcomings of our ability to deal with it.First of all, we implemented a standard TCD setup providing a long incubation period to represent the nature of the epidemic within one population possessing a specific structure.

Formulation of TCDIL 2 IL 12 Z model
Here, a new mathematical model is developed for lung cancer by introducing IL 2 , IL 12 cytokines, and anti-PD- L1 inhibitor for treatment, whereas the TCD approach was utilized in the current model.The current model for examining cancer disease was released in June 2023 and is given in 45 .TCDIL 2 IL 12 Z is the name given to this novel model, in which "T" stands for tumor cells, "C" for CD8+ T cells, "D" for dendritic cells, " IL 2 " & " IL 12 " for cytokines, and "Z" for anti-PD-L1 inhibitor.
We present a number of crucial parameters in this model: The tumor's logistic growth is described by the expression " αT(1 − βT) ", " γ " indicates the constant rate at which dendritic cells destroy tumor cells, " φ " indi- cates the rate at which CD8+ T cells eliminate tumor cells, " κ " indicates the rate at which CD8+ T cells naturally die, " µ " characterizes the sources that produce dendritic cells, " ρ " signifies the rate at which CD8+ T cells render dendritic cells inactive, " ω " indicates the rate at which dendritic cells naturally die, " " represents the source of IL 2 to reduce the dendritic cell's, "d" and " ψ " denoting the rate at which these cells boost the immune system through the activity of CD8+ T cells and "a" denoting the anti-PD-L1 inhibitor's natural death rate.
Figure 1 shows flow diagram for newly developed TCD IL 2 IL 12 Z model.The model that was created using the anti-PD-L1 inhibitor effect and generalized hypothesis involving cytokines is shown below:

Qualitative and quantitative analysis
Within this section, I conduct a comprehensive analysis of equilibrium points.To determine these points, it is necessary to equate the left-hand side of the system (3) to 0.
The equilibrium point corresponding to the absence of disease in this model is ( For the recently created system employing the next generation approach, the reproductive number is

Sensitivity analysis
Sensitivity analysis is useful for evaluating the relative influence of several factors on the stability of a model, especially when dealing with ambiguous data.Moreover, this research helps identify the critical process factors.Reproductive no."R ′′ 0 is The sensitivity of R 0 can be examined by computing the partial derivatives of the threshold with respect to the pertinent parameters in the manner described below: It is evident that the value of R 0 is quite sensitive when we change the settings.We find that in our analysis, the parameters ρ expand while µ, κ, , ω shrink.As a result, for efficient infection management, prevention should come before therapy.As shown in Fig. 2, the previously mentioned indices aid in determining the critical elements that determine the infection's potential for propagation.

Local stability analysis of equilibrium points
The local stability of equilibria is described here, along with related proofs.

Theorem 1
The proposed fractional-order lung cancer model's disease-free equilibrium point exhibits local asymptotic stability when R 0 is less than 1, but becomes unstable when R 0 is greater than 1.
Proof Assume the Jacobian matrix, abbreviated as 'J' as follows to examine the stability of the model at the points E 0 : .
. Consequently, all of the eigenvalues have negative real parts, indicating the local asymptotic stability of the system.

Global stability for developed system
We investigate global stability and find the criteria for illness elimination using Lyapunov's method and LaSalle's idea of invariance.

Lyapunov's first derivative
Theorem 2 If R 0 > 1, then the lung cancer model's endemismic equilibrium is globally asymptotically stable.Proof The expression for the Lyapunov function is as follows.
By using a derivative on both sides, we get We get, we can write where Vol:.( 1234567890) t L < 0 is obtained; on the other hand, when As we can see, then the D 2 is globally uniformly stable in Ŵ , in accordance with Lasalles' idea of invariance.

Chaos control
System (3) is a fractional-order system with a controlled design that is stabilized based on its points of equilibrium through the use of the linear feedback regulate technique.
where the system's equilibrium is represented by D 1 and the control parameters are ω 1 , ω 2 , ω 3 , ω 4 , ω 5 , and ω 6 .At D 1 , the related Jacobian matrix is found as follows: At D 1

Thus, the characteristics equation is
The following are the eigenvalues (�) that result from solving the aforementioned determinant: Vol.:(0123456789) Thus, the characteristics equation is The following are the eigenvalues (�) that result from solving the aforementioned determinant: As a result, all of the eigenvalues bear negative real parts and this proves that it is locally asymptotically stable.
In each of the above equated is eigenvalue, negative real number or a complex number with negative real parts therefore, for any 0 < q < 5 , equilibrium points D 1 , D 2 are asymptotic stability.

Bounded and positive solutions
We illustrate the derived model's positivity and boundedness in this section.

Theorem 3 The initial values under consideration as:
Consequently, {T, C, D, IL 2 , IL 12 , Z} will have positive solutions, with ∀ t ≥.
Proof To demonstrate the higher caliber of the answers, we shall start the main analysis.These methods have good effects and successfully handle problems in the actual world.The methods described in references [46][47][48] will be employed by us.We'll look at the prerequisites in this section to make sure the recently created model produces results that are favorable.We shall set the standard in order to achieve this: here " D η " represents the η domain.Now, for T(t): we get, here the time component is denoted by "b ′′ .This proves that ∀ t ≥ 0, T(t) must be positive.Now, for C(t): = −1.0486= −0.5238= −3.2400= −4.003= −5.003= −6.0400.Proof As can be seen from the above theorem, methods are detailed in 49 .The answers of our constructed model must be positive ∀ t ≥ 0 .Given that X = T + C + D .so provided as follows.
We get, Additionally, it has X υ = IL 2 + IL 12 + Z .We have so evolved.
After calculating t → ∞ and solving the preceding equation, we obtain The mathematical solutions (3) for the model are limited to .
This proves that all solutions in domain stay positive and consistent with given beginning conditions for any t ≥ 0.
Theorem 5 In addition to the beginning condition, the recently created lung cancer model Eq.(3) in R 6 + is unique and positive invariant.
Proof In this specific case, we used the process outlined in 49 .We possess Equation ( 4) states that if (T 0 , C 0 , D 0 , IL 0 2 , IL 0 12 , Z 0 ) ∈ R 6 + , then our obtained solution cannot escape from the hyperplane.This demonstrates that the domain R 6 + becomes a positive invariant.

Impact of global derivatives for existence and uniqueness of solution
The most often used integral in the literature is generally acknowledged to be the Riemann-Stieltjes integral.If

Riemann-Stieltjes integral is:
The global derivative of y(x) with respect to n(x) is When the numerator and denominator of the aforementioned functions differentiate, we obtain Considering the following: n ′ (x) � = 0, ∀x ∈ D n ′ .Using the global derivative in place of the conventional deriva- tive, we will now examine the effect on the lung cancer.
We shall assume that n is differentiable for the sake of cleanliness. ( Y n (x) = y(x)dn(x), www.nature.com/scientificreports/An proper selection of the function n(t) will result in a certain result.Fractal movement will be seen, for example, if n(t) = t η , where η is a real value.The conditions that required us to act were The system's unique solution is shown in the example below for the created system. where The first two conditions are as follows, which we must verify.
for condition: where for condition: where Vol where The linear growth criterion is therefore met.We also confirm the Lipschitz criterion in the following way.If where where If involving Thus, with the following condition, there exists a unique solution for the system (3).

Solutions by fractal fractional operator
The newly developed model, stated in Eq. ( 3), will now have a numerical solution developed for it.Instead of using a traditional derivative operator in this instance, we employ an ML kernel.
For clarity, where The system of Eq. ( 4) yields the following when the Newton polynomial is substituted.

Simulation explanation
The theoretical results and their efficacy are investigated using the advanced approach.We do a simulation analysis of the recently created system TCDIL 2 IL 12 Z.Using non-integer parametric variables in the lung cancer model has allowed us to arrive at some fascinating results.We may get dependable answers by decreasing fractional values for the individuals T(t), C(t), D(t), IL 2 (t), IL 12 (t) , and Z(t) in Figs. 3, 4, 5, 6, 7 and 8. Approximate answers for the lung cancer model are found using MATLAB code.T(0) = 1.0,C(0) = 0.8, D(0) = 0.3, IL 2 (0) = 0.4, IL 12 (0) = 0.4 , and Z(0) = 0.3 are the starting conditions employed in the recently constructed model.The parameters that were employed in the system that was built Vol:.( 1234567890 The dynamics of tumor cell T, cancer cell C, IL 2 cytokine, IL 12 cytokine, and Z anti-PD-L1 inhibitor are shown in Figures 3, 4, 6, 7, and 8.During this time, all of the compartments sharply decrease before approaching their stable positions using various dimensions.The dynamics of dendritic cells D are shown in Fig. 5, where the number of individuals grows fast and, with the passage of time, each compartment achieves its stable location utilizing a distinct dimension.Cytokines and anti-PD-L1 inhibitors are seen to cause a dramatic drop in cancer cells; this can be seen in Figs. 4, 6, 7, and 8 using various dimensions.A comparison of dimensions 0.2 and 0.5 shows similar results with little effects; however, reducing dimensions yields more acceptable results, as shown in Fig. 8. Furthermore, anti-PD-L1 cells and cytokines have been shown to support the immune system by boosting the generation of CD4+T and CD8+T lymphocytes and reducing the number of cancer cells.Along with aiding in the creation of cells that the body's cancer cells kill, anti-PD-L1 also aids in the reduction of cancerous cells.It foretells what this study will uncover in the future and how we will be able to more effectively lower the amount of cancer cells that spread throughout the body.It can be deduced from Figs. 5 and 6 that the combine measures of dendritic cells and cytokine IL2 need to increase as it play the key role in the activation of antigen-specific CD8+T lymphocytes to improve the low immune individuals which is initiative step to fight against cancer cells while cytokine IL2 directly boost the immune system.Similarly, the anti-PD-L1 inhibitor rises gradually which directly helps to kill PD-L1 cells and cancer cells.It provide us better control by taking combine measures to boost immune system and killing lung cancer cells in the body which can be seen in Figs. 5,   observed that, when we compare to the traditional derivative, the FFM technique produces superior results for every sub-compartment by decreasing fractional values and its dimensions.It is also mentioned that as dimensions and fractional values are decreased, the solutions for every compartment become more reliable and exact for control purpose.

Conclusion
This paper formulates a fractional order TCDIL 2 IL 12 Z lung cancer model by adding cytokines and an anti-PD-L1 inhibitor to strengthen the immune system in those with weakened immune systems.By injecting anti-cancer cells, which strengthen people's immune systems and eradicate sickness from the environment, we show how to prevent the spread of illness.The harmful lung cancer illness is studied along with prevention and therapy options to assess the true global effect of lung cancer.To do this, a quantitative and qualitative investigation of the generated system is evaluated to confirm its stable state for a continuous dynamical system.In order to understand the dynamics of the epidemic, it is necessary to know how the model behaves under constrained conditions, which is determined by local stability analysis.We also confirm that there exist bounded and unique solutions for the fractional order lung cancer disease model.We confirm its existence and analyze the effects of international efforts to slow the lung cancer disease's spread.In order to evaluate the overall effect of cytokines and anti-PD-L1 inhibitor for persons with weakened immune systems, the model is examined for global stability using Lyapunov functions with first derivative.It has been demonstrated that cytokines and anti-PD-L1 inhibitor therapies for patients with compromised immune systems reduce the number of cancer cases.Using various fractional values and dependable, practical results, the Fractal-Fractional Operator (FFO) is utilized to continuously track the   www.nature.com/scientificreports/disease's progress.After including cytokines and anti-PD-L1 inhibitor measures, we use MATLAB for numerical simulation to observe the control of the lung cancer disease in the community.Predictions for future research will help to comprehend the behavior and environmental spread of lung cancer sickness as well as the early detection process may also be made based on our well-founded findings.In future the developed lung cancer model can be tested on any specific reginal real data and will be helpful control strategies.Also optimal control strategies can also be applied, it may provide better control under modified fractional operator.As its newly developed system, so fuzzy and optimization approach can be unutilized to investigate lung cancer in different aspects.

Figure 1 .
Figure 1.The model formulation is shown in the flow chart.

Figure 2 .
Figure 2. Reproductive number behavior for the newly developed system under different parameter effects.

Figure 4 .
Figure 4. Simulation of C(t) using FFM with different dimensions.

Figure 5 .
Figure 5. Simulation of D(t) using FFM with different dimensions.

Figure 6 .
Figure 6.Simulation of IL 2 (t) using FFM with different dimensions.